Maxwell Strata and Cut Locus in Sub-Riemannian Problem on Engel group

TL;DR

This paper analyzes the global structure of geodesics, cut locus, and Maxwell sets in the sub-Riemannian Engel group, providing detailed stratification and symmetry descriptions relevant for optimal control and geometric analysis.

Contribution

It offers a comprehensive description of the cut locus, Maxwell set, and conjugate points in the Engel group, including stratification and symmetry properties, for the first time.

Findings

The cut locus has a stratification with 6 three-dimensional, 12 two-dimensional, and 2 one-dimensional strata.

Maxwell strata of multiplicity 2 are three-dimensional, conjugate points form two-dimensional strata, and infinite multiplicity Maxwell strata are one-dimensional.

Projections of geodesics are Euler elasticae, and the optimal synthesis is explicitly described.

Abstract

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimen-sional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i.e., the set of minimizers for each terminal point in the Engel group.